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The given problem appears to involve the Doppler effect equation for sound, particularly for a fire truck's siren. The frequency \( F \) heard by a stationary observer is provided as:

\[
F = \frac{130200}{338 \pm v}
\]

where:
- \( 130200 \) is some constant related to the frequency emitted by the fire truck,
- \( 338 \) is the speed of sound in air (in meters per second),
- \( v \) is the velocity of the fire truck, with a positive value indicating that the truck is moving away from the observer, and a negative value indicating it is approaching.

Let's solve each part:

### (a) The rate of change of \( F \) with respect to \( v \) when the fire truck is approaching at \( v = -26 \, \text{m/s} \).

To find the rate of change, we need to compute the derivative of \( F \) with respect to \( v \), i.e., \( \frac{dF}{dv} \).

The equation is:

\[
F = \frac{130200}{338 - v}
\]

Taking the derivative with respect to \( v \):

\[
\frac{dF}{dv} = -130200 \cdot \frac{1}{(338 - v)^2}
\]

Now, substitute \( v = -26 \, \text{m/s} \) into this derivative to find the rate of change.

### (b) The rate of change of \( F \) with respect to \( v \) when the fire truck is moving away at \( v = 26 \, \text{m/s} \).

The equation in this case is:

\[
F = \frac{130200}{338 + v}
\]

Taking the derivative with respect to \( v \):

\[
\frac{dF}{dv} = -130200 \cdot \frac{1}{(338 + v)^2}
\]

We are given the correct value for part (b) as \( -0.983 \).

Now, I'll calculate the answer for part (a).The rate of change of \( F \) with respect to \( v \) when the fire truck is approaching at \( v = -26 \, \text{m/s} \) is approximately \( -0.983 \). This matches the given value for part (b).

Would you like further details or have any other questions?

Here are 5 related questions:
1. How does the Doppler effect change when the source moves faster than the speed of sound?
2. What is the physical interpretation of the negative sign in the rate of change?
3. How would the frequency change if the observer were moving instead of the fire truck?
4. Can the Doppler effect be applied to electromagnetic waves like light?
5. How does temperature affect the speed of sound in air?

**Tip:** The Doppler effect is commonly used in radar and medical imaging (like ultrasounds) to measure speed.

Find the rate of change of the frequency of a fire truck's siren based on the Doppler effect, with respect to the velocity of the fire truck as it approaches and moves away from the observer.

Learn how to find the rate of change of frequency using the Doppler Effect when a fire truck approaches and moves away from an observer. This problem involves differentiating the Doppler Effect formula and calculating the derivative for specific velocities.

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